Theory of Variation

In this chapter, we review some basic concepts of deterministic integration theory in the sense of Riemann-Stieltjes and Lebesgue-Stieltjes. We shall see that, unfortunately, the trajectories of a Brownian motion (and, in general, of a martingale) do not have sufficient regularity to use such theories to define the Brownian integral in a deterministic sense, path by path. To understand this fact, it is necessary to introduce the concepts of first and second (or quadratic) variation of a function, which are crucial in the construction of the stochastic integral. In the second part of the chapter, we introduce an important class of stochastic processes called semimartingales. A semimartingale is the sum of a local martingale with a process whose trajectories are of bounded variation: under appropriate assumptions, such decomposition is unique.